# Full text of "The Statistical Method in Problems of Water Supply Quality"

## See other formats

```STOP

Early Journal Content on JSTOR, Free to Anyone in the World

the world by JSTOR.

Known as the Early Journal Content, this set of works include research articles, news, letters, and other
writings published in more than 200 of the oldest leading academic journals. The works date from the
mid-seventeenth to the early twentieth centuries.

We encourage people to read and share the Early Journal Content openly and to tell others that this
resource exists. People may post this content online or redistribute in any way for non-commercial
purposes.

journal-content .

JSTOR is a digital library of academic journals, books, and primary source objects. JSTOR helps people
discover, use, and build upon a wide range of content through a powerful research and teaching
platform, and preserves this content for future generations. JSTOR is part of ITHAKA, a not-for-profit
contact support@jstor.org.

188 American Statistical Association [48

THE STATISTICAL METHOD IN PROBLEMS OF WATER
SUPPLY QUALITY

By Abel Wolman, Maryland State Department of Health

INTRODUCTORY

The concept of water supply quality has the simplicity of the
unknown to the layman, but the complexity of the universe to the
sanitarian. If one uses the mathematician's measure of the complexity
of a function — the number of its attributes — the problem of water
supply quality, a function dependent upon mutually active natural,
physical, chemical and biological phenomena, offers an attractive
field of study to the statistician. For the professional statistician has
been concerned always with "quantitative data affected by a multiplic-
ity of causes" 1 and with their elucidation. In considering the causes
operating to produce relatively good or bad waters, such as rainfalls,
pollution, purification, etc., and their interpretation upon the basis of
laboratory findings and personal surveys it becomes manifest that
problems of water supply quality fall well within the scope of statistical
method. Just as in all statistical problems, so in that of water supply
quality, the investigator is confronted with the two-fold task of deter-
mining the method of evaluating the units of interpretation and of
defining the limiting values of such units. The method of approach
to each problem involves a statistical viewpoint, as well as a quantita-
tive methodology. The present paper has been prepared in order to
illustrate, in as brief terms as possible, this statistical method of ap-
proach, by developing therein a few examples of its application to the
question of water supply quality. The writer plans to trace the
evolution of the concept of water supply quality in the sanitarian's
mind and to point out in such a development the function which the
statistical art has performed or may be expected to supply in the
future. The discussion appears to be a necessary one since hitherto
the water supply investigator has been accused of an aversion for the
quantitative sciences, while, on the other hand, the professional statis-
tician has shown a neglect of a field which perhaps did not appear to
be worthy of his mettle. The present study may serve to remove this
friendly distrust which retards in a degree progress in critical studies
of water supply quality.

I. THE LABORATORY EXAMINATION OF WATER SUPPLIES

The sanitary quality of water supply must be predicated necessarily
upon the demonstration of its relative inability to produce disease.

49] Statistical Method in Problems of Water Supply Quality 189

With the present germ theory of disease, such a demonstration resolves
itself into the laboratory problem of enumerating the number and
types of pathogenic organisms in stated quantities of water. It is
apparent, therefore, that the technique in this instance is largely
bacteriological, and the discussion, for purposes of simplicity, may be
restricted to the problems of the evaluation of bacterial units, as illus-
trative of the statistical method.

It is manifestly impossible and impracticable to examine an un-
known water in such manner as to determine its content of all kinds of
bacteria or even of those relatively few classes of specific organisms
which it is known are both disease-producing and capable of living in
water. It is more desirable as well as convenient, therefore, to choose
one family or group of micro-organisms whose natural habitat and
life history are similar to the variety of pathogenic organisms and
whose detection by laboratory methods is most simple and speedy.
Bacteriologists have concluded that a particular class of bacteria
serves as the most convenient index to water supply quality or con-
tamination. They have chosen as this index class or type the so-called
colon or bacillus coli group. The B. coli group has been so selected,
because its origin is in general the colon or digestive tract of man and
its presence is usually indicative of human sewage pollution (the
possible and probable existence of colon types in other environments
need not concern us at this point).

One of the primary objects of the bacteriologist, therefore, is to
differentiate the bacterial species present in a water supply, so as to
demonstrate the presence or absence of members of the B. coli group.
In addition, it is necessary to obtain some idea of the relative frequency
of such a group, since smaller numbers naturally connote a more
remote pollution, due to the dying off of bacteria in the unfavorable
environment of water, to the presence of antagonistic life, and to other
natural and artificial barriers to its development. The problems
arising in the laboratory differentiation of bacterial varieties offers,
therefore, material for an initial example. Two general methods of
distinguishing groups of bacteria are available. Both are based upon
the method of differences. In the one case, morphological or structural
characteristics, and in the other, metabolic distinctions control.
Various classifications of the colon group, for instance, are based upon
its ability to produce acid and gas from fermentable substances.
Investigators have observed that certain types of B. coli ferment
such complex organic compounds as sucrose, dulcitol and raffinose
while others do not. Differences in the amount and character of gas
formation from certain substances distinguish other types of bacteria.

190 American Statistical Association [50

In all classifications, however, it has been recognized that the same
group may have a variety of reactions which overlap partially those
of other groups. Two types of bacteria, for instance, may both fer-
ment sucrose, but may differ in their effect upon a second or third
compound. This gives rise naturally to a vast amount of possible
combinations between characters and Levine 2 points out that "as
the number of fermentable substances increases, the number of varie-
ties increases geometrically approaching infinity. The number of
'varieties' is given by the formula 2" where V is the number of char-
acters studied. Thus with 8 characters there are 256 possible com-
binations; this number rises to 1,024 with 10 characters and to 65,536
when 16 characters are observed. The absurdity of regarding each
character as of similar and equal differential value is thus evident."

Levine, as well as other more recent investigators, has concluded
that the principle of the correlation of characters should be emphasized
in the attempt to distinguish bacterial species. He points out that
certain properties have been universally accepted, after long checking,
as reliable evidences of bacterial differences. Among such properties,
he enumerates the selective dyeing of bacteria, their powers of spore
formation, and their adaptation to aerobic or anaerobic development.
The taxonomic value of the characters of motility, indol formation,
and fermentation of certain compounds, on the other hand, he assumes
to be still debatable. In order to avoid the adoption of a confusing
classification of bacteria upon the basis of every character studied (of
which we have indicated only a few) he has recourse to a basis of sub-
division "on that character which gives the greatest amount of infor-
mation as to the manner in which the resulting sub-groups react with
respect to other characters. " 2 By making use of the above principle
Levine evolves a classification of coli-like bacteria which is based almost
completely upon statistically evaluated correlated characters. For the
purpose of this study, he recognizes two main strains of bacteria, the
B. coli and the B. aerogenes-cloacae group, which earlier investigations
have shown to be distinguishable most often by their reactions to
methyl-red and to the Voges-Proskauer reagent. The first strain is
usually methyl-red positive and Voges-Proskauer negative, while the
second strain shows the reverse. The justification of this initial sub-
division into two main groups consists in the fact that the strains thus
subdivided show end products of carbohydrate fermentations of two
entirely distinct kinds.

Levine's procedure consists in tabulating all of the reactions of the
organisms studied in each of the above two groups in two different
tables, from which are calculated the coefficients of correlation for each

51] Statistical Method in Problems of Water Supply Quality 191

pair of characters. He selects, then, for subdivision that character
which gives the highest coefficient of correlation with the greatest
number of other characters. For these resulting sub-groups new corre-
lation tables are prepared and further subdivision is made. These
sub-groups are regarded as species and each is assigned its name.

In order to illustrate Levine's use of the coefficient of correlation for
taxonomic purposes, let us follow his procedure in the subdivision of
the B. coli, or methy-red positive and Voges-Proskauer negative,
group of bacteria. For the 182 strains of this group that were studied
by means of microscopic and metabolic methods, the coefficients of
correlation shown in Table I were obtained.

TABLE I.
COEFFICIENTS OF CORRELATION OBTAINED FROM PAIRS OF CHARACTERS
AMONG 182 STRAINS OF THE B. COLI GROUP

Motility

Indol

Sucrose

Raffinose

Dulcitol

Glycerol

— .39

+ .53

+ .43

+ .53

+ .18

— .39

+ .08

+ .00

+ .02

— .28

+ .53

+ .08

+ .99

+ .58

— .38

+ .43

+ .00

+ .99

+ .58

— .29

+ .53

+ .02

+ .5S

+ .58

— .21

+ .18

-.28

— .38

— .29

— .21

+ .40

+ .76

+ .20

+ .27

+ .60

+ .52

Salicin

Motility
Indol . . .
Sucrose . .
Raffinose
Dulcitol .
Glycerol
Salicin . .

+ .40
+ .76
+ .20
+ .27
+ .60
+ .52

Since Levine's criterion for the choice of a character for subdivision
is that that character should give the highest coefficient of correlation
with most other characters, it is apparent, from an inspection of Table
I, that sucrose, raffinose, dulcitol, and salicin meet this criterion more
completely than do other properties. For special technical reasons,
Levine chooses sucrose for primary division of the B. coli group and
obtains by differentiation on sucrose ninety-three strains of the sucrose
positive and eighty-nine strains of the sucrose negative groups. These
two groups combined form, of course, the total of 182 strains initially
chosen for study. Further study of the sucrose positive strains dis-
closes a series of coefficients of correlation of characters as shown in
Table II.

TABLE II.
COEFFICIENTS OF CORRELATION FOR PAIRS OF CHARACTERS AMONG 93 SU-
CROSE POSITIVE STRAINS OF THE B. COLI GROUP

Motility

Indol

Dulcitol

Glycerol

Salicin

Motility

Indol

Dulcitol

Glycerol

— .27
+ .67
+ .40
+ .54

— .27

+ .05

— .42
+ .28

+ .67
+ .05

— .32
+ .39

+ .40

— .42

— .32

+ .32

+ .54
+ .28
+ .39
+ .32

192 American Statistical Association [52

Table II indicates that motility is the best correlated character and
this property provides, therefore, for two further sub-groups, a sucrose-
positive motile sub-group and a sucrose-positive non-motile group.
These sub-groups are treated in the manner already illustrated ahd the
coefficient of correlation for different characters provide for further
subdivision. With the aid of this statistical interpretation of his stud-
ies of 333 coli-like bacteria, isolated from various sources, Levine sug-
gests a classification of bacterial varieties. The summary of this
classification need not be repeated here, since the reader is interested
more in his method of attack than in the resulting bacteriological
findings.

Such classifications as Levine's supply the sanitarian with the quali-
tative information necessary for the interpretation of one phase of the
water supply quality problem.* The analyst dealing with waters is
concerned not only with the nature of the bacterial types present
therein, but also in the magnitude of their content, since it is the latter
which indicates the degree and the remoteness of pollution. In the
search for a potable water, it is often useless to seek that water which
has no possible source of contamination, but it is always necessary to
determine the quantitative bacterial importance of the latter. The
methods so far described answer only one question, that is, what types
of bacteria are present in the water. In the solution of the second
inquiry, regarding the number of a particular type in a stated quantity
of water, statistical method has played recently an important part.

In the simpler tests for the B; coli group in waters, the so-called
fermentation-tubes are used. These tubes contain the medium
selected for most efficient differentiation of the B. coli group from other
kinds of bacteria and are inoculated with specific quantities of the water
to be tested. The production of gas in the tubes after stated periods
of incubation indicates the presence of the B. coli group. Our knowl-
edge that of five tubes, each inoculated with 0.1 c.c. of the water, four
show the presence of the organism, is of value, but more important is
the additional fact that such a series of findings indicates that the prob-
able number of organisms in the sample tested is about 1,600 per 100c. c.
This conversion of qualitative fermentation-tube results into quanti-
tative values is of special interest to the statistician.

In 1915, McCrady 3 showed that "the frequency of the appearance of
the fermenting organism in the volume drawn from the sample for the
test is an exponential function of the number of such organisms in the
sample," and that "every fermentation-tube result, whether simple or
compound, corresponds to one most probable number of organisms."

*The subdivisions Levine develops have their importance to the investigator in the fact that species
or varieties appear to be somewhat correlated with habitat or source of pollution.

53] Statistical Method in Problems of Water Supply Quality 193

By employing the theory of probabilities, he demonstrates that, given

p
the result " in 1 volume," for instance, the corresponding most

P + Q

probable number is given by the solution for x of the equation

\ V I p+q

Thus, for the result "five out of ten tubes positive in 1 c.c," the most

probable number is given by solution of the equation 1 — .99* = 5/10,

since V — 100 c.c, assumed as the original quantity of water sampled.

The equation being solved, a; = 69 or the most probable number of B.

coli in the sample, per 100 c.c.

For compound results, such as — - — - in 10 c.c, in 1 c.c, a

p+q r+s

more complicated formula is employed which is built up, as follows: 3
For the result in 10 c.c. the equation becomes (p+q) (log . 9)

= — which is obtained by differentiating for a maximum the

1 - . 9*

equation given in the earlier paragraph for the probability of the

results.

7) r
If the result is in 10 c.c, in 1 c.c, the equation stands

p+q r+s

(p+q) (log .9) + (r+s)(log ■")= i_ o* 1-99*

where (p+q) = number of tubes inoculated with 10 c.c. of sample
(r+s) = number of tubes inoculated with 1 c.c of sample

x = number of fermenting organisms in 100 c.c. of sample

p and r = number of tubes giving positive results in 10 and 1 c.c.

respectively.

If lower additional quantities of water are tested, extra similar

terms are added to each side of the above equation. This equation

has been modified by Wolman and Weaver 4 into

lOOp lOr

100(p+g) + 10(r+s) =

1-.9* l-.99 x

since, approximately, log .9 = 10 log .99 = 100 log .999.

McCrady published later 5 a series of tables for the rapid interpre-
tation of these results which makes the standardized use of the prob-
able numbers of B. coli possible for the water supply investigator.*

*The assumption of McCrady that the distribution of B. coli is similar to that in a mixture of a few
red balls with many white balls is to be contrasted with the hypothesis of other workers that bacteria
are uniformly distributed in water (G. C. Whipple 13 ). More recent independent investigators, however,

194 American Statistical Association [54

The work of McCrady was followed by other investigations dealing
with the numerical interpretations of B. coli tests, of which the more
important are Stein 6 , 7 , 8 , Greenwood and Yule 9 , and Wells 10 , ", I2
The results of Stein and Greenwood and Yule, although differing in
technique and in additional interesting viewpoints, are in substantial
able interesting statistical material to the B. coli problem by introduc-
ing the so-called B. coli factor method, in which he considers the most
probable number of B. coli per c.c. from the percentage of positive
tests, the expected error of results, the study of the distribution of coli
during a series of tests, and the "coli characteristic" which attempts to
show by one figure, the average coli, the expected error and the variable
distribution.

The discussion of the problem by Greenwood and Yule 9 has all the
intricacy and mathematical complexity usually associated with Yule's
contributions. Their findings, however, agree with those of McCrady
and Stein. Greenwood and Yule, for instance, give as their formula
for the number of B. coli per c.c, when using several tubes with 1 c.c.

each tj v o o i V+Q
x = ii. con per c.c. =2.3 log

q

whereas McCrady gives for the same condition (using an original size
sample of 1,000 c.c.)

log — — log — — log — - —

x = P+q = 2+1 = ?+g = -2.3 1o g q

1 ,000 log .999 - 1 ,000 ( . 0004344) - . 4344 P + q

= 2.3 log 2+1

q

Perhaps the mathematician's interest may be aroused to the sani-
tarian's problems of water supply by the mere examination of Green-
wood and Yule's discussions, while the bacteriologist may view with
some alarm the same paper. It should be postulated in either case,
however, that superficial considerations should not prevent the mutual
aid which these two branches of science may extend to each other.
While such complexity of treatment of the numerical interpretation of
fermentation-tube tests as is indicated by the formula

\ mi / \m2

,_/:[<

hmtul i p—hm \ _ p—hamil i — hai

/hi T / \m\ I \mi

"Jl-e- ha nY n ] (

55] Statistical Method in Problems of Water Stipply Quality 195

may attract the statistician, it is hoped that it may not at the same
time deter the laboratory technician from the adoption of devices
which provide for more adequate solutions of his problems. Emphasis
must be placed upon the fact that the mental attitude resulting from
the adoption of statistical method has much promise in a field of
endeavor where laboratory findings are too infrequently tested for
accuracy of interpretation and rarely treated as examples of mass
phenomena. The work of such men as Stein and McCrady has done
much to introduce such methods by clarifying our concepts of fermen-
tation tube results and their relative significance.

That the statistical method is an important asset in the exposition
of laboratory findings is illustrated in another series of studies of various
phases of water supply. Whipple 13 , for instance, has demonstrated
that "if, in a series of daily observations of the number of bacteria in
a filter effluent extending over a year the deviation of any determination
from the mean should be found to be more than five times as much as
the probable error, to use a round number, this should be rejected from
the series as being, for some reason or other, abnormal." He has
made important contributions to the study of the frequency distribu-
tions of measures of various bacteriological, biological, and chemical
characteristics of water, such as the preliminary finding that extended
series of filter effluent results follow definite statistical laws in their
distribution. His conclusion has been further substantiated by the
more recent study of Wolman 14 of thousands of laboratory findings,
in which it is indicated that the logarithms of bacterial counts, through
long periods of time, have the characteristic normal probability dis-
tribution of more familiar biological statistical data.

It is of considerable interest to refer at this point to a form of graph
presentation of data developed by a sanitary engineer which may be
unfamiliar to most statisticians. Allen Hazen 16 in 1914 devised a
form of chart ruled with a horizontal scale so divided tha the curve of
probability would plot thereon as a straight line. Any series of ob-
servations, therefore, which varied in accordance with the probability
law would plot also as a straight line. Illustrations of the use of such
paper in water supply problems may be found in the original paper of
Hazen 16 and in subsequent discussions by Whipple 13 and Wolman 14 .

Stein 16 , in his study of the bacterial count in water and sewage, has
added considerable material to our conceptions of the variability of
laboratory findings and their importance in practical studies. He has
concluded, after an interesting detailed analysis of the problem, that :

(a) For platings of a single sample of water, the mean error is equal
to the square root of the number of colonies on a single plate, or the
square root of the average number of colonies on several plates.

196 American Statistical Association [56

(b) The variations to be expected for careful and accurate work with
bacterial counts are indicated by :

(1) Standard Deviation of ± 12%

(2) Deviation (1 in 10 times) of ± 25%
For ordinary routine work:

(1) Standard Deviation of ± 25%

(2) Deviation (1 in 10 times) of ± 50%

His comparison of the characteristics of bacteriological data with
certain mathematical series should be of interest to the reader, since he
shows, for example, that for daily tests of Lake Erie water for one
month the Lexian Ratio is 29.00 and the Disturbancy Coefficient 124.00,
while the corresponding values for a normal mathematical series (Ber-
nouilli) are given as 1.00 and 0.00 respectively.

II. THE INTERPRETATION OF THE QUALITY OF WATER SUPPLIES

In preceding paragraphs the writer has indicated a few of the problems
encountered in the laboratory technique of water supply examination,
which lend themselves to statistical treatment. It has been impossible
to include in the present brief paper any complete survey of such appli-
cations to other phases of laboratory procedure, but sufficient material
has been presented, to demonstrate that the data in the field of labora-
tory technique have considerable to offer to the professional statistician
as bases for the development of interpretative principles of quality.

The writer believes that some mention should be made briefly of
certain interesting possibilities of development in the application of
statistical method to general problems of laboratory procedure. The
use of the coefficient of partial correlation, for instance, does not appear
to have been introduced widely in the interpretation of laboratory find-
ings, yet the necessity for its application is most apparent. Often
investigative work in water supplies is carried out on a large or plant
scale with the aid of analytical laboratory methods. In the study of
the chlorination of a water supply, for example, a number of different
variable quantities such as turbidity, color, organic content, and bac-
terial densities have their effect in modifying the efficiency of the dis-
infection process. In practically all conclusions from such studies no
attempt is made to determine mathematically the effect of such varia-
bles, other than by mere inspection of tabulated data. There is little
doubt that erroneous conclusions are often obtained through the failure
to evaluate quantitatively the importance of fluctuations in the various
characteristics of waters subject to chlorination. It is almost impossi-
ble to determine by qualitative inspection of a series of daily observa-
tions, over an entire year, of temperature, turbidity, color, organic

57] Statistical Method in Problems of Water Supply Quality 197

content, and bacterial density in a water supply, whether the effect of
a constant dosage of chlorine is influenced more greatly by any one of
the above characteristics or by a combination of several or all of them.

The same problem arises, of course, in the study of any of the phe-
nomena associated with the purification of water supplies. In the co-
agulation of suspended matter in water, for instance, all the variables
such as time, agitation, temperature, hydrogen-ion concentration,
nature of suspended matter, and character of coagulant play an inter-
connected part. The principle of partial correlation could be adapted
with profit to these problems of associated phenomena.

The application of such a statistical principle as pointed out above
is complicated, however, by the fact that the more simple statistical
coefficients usually cannot be directly applied to the problems encoun-
tered, on account of the fact that such measures presuppose the use of
data having a symmetrical or Gaussian distribution, while the phe-
nomena with which the sanitarian has to deal often are characterized
by asymmetrical distributions. 17 , 18

Michael 18 has discussed in this connection the determination of the
most probable number of bacteria present in a sample and has demon-
strated that it is not permissible to apply the probable error in the usual
manner on account of the fact that the logarithms of the plate counts,
and not the counts themselves, show a Gaussian frequency distribu-
tion 19 . McEwen and Michael 17 in another field of investigation have
been confronted with the same problem of determining the "functional
relation of one variable to each of a number of correlated variables"
where such variables do not show the usual symmetrical frequency
distribution. It is manifestly impossible to extend in this paper the
elucidation of these applications of statistical method to problems of
laboratory and plant, but the reader may find profitable data in the

The opportunity for the application of statistical tests to problems of
water supply quality is not restricted, however, to the materials of the
analyst. The consideration of the potability of a supply involves
always a series of mutually active attributes, each of which has its im-
portance in determining the character of the water. The concept of
quality connotes, therefore, a composite of properly weighted individ-
ual and fundamental units, in the evaluation of which statistics again
comes to the fore.

It is unfortunate, however, that in the field of interpretation of
quality statistical method has been even slower of application than in
the corresponding study of laboratory data. The quantitative eval-
uation of sanitary data has always given way to the liberal exercise of

198 American Statistical Association [58

expert personal judgment. Where a multiplicity of causes predeter-
mines a phenomenon, such as quality, it was thought that a proper
perspective was possible only through the development of a maturity
of judgment in which the play of the manifold effects was qualitatively
summarized rather than quantitatively analyzed. As the methods of
diagnosis of quality developed, however, the opportunity for the fruit-
ful application of the principles of mass phenomena gradually becomes
apparent. With this development of a new viewpoint, good as well as
evil sometimes resulted. A complete swinging of the pendulum to the
quantitative side of interpretation was feared, where the attempt was
made to substitute for individual experience and judgment pseudo-
quantitative measures of doubtful significance. Some of these efforts,
in which statistical laws frequently were ignored, will be discussed later
in this paper. In general, however, a realization is gradually coming
over the sanitarian that statistics as a means, rather than as an end,
has much to offer in the clarification of his problems. If the succeeding
pages seem somewhat bare, in their statistical implication, the pro-
fessional statistician should remember that the concepts there dis-
cussed mark the advance of a new light in sanitary engineering, which,
though feeble in its flicker, gives promise of a greater brilliance in the
not distant future.

Attempts to formulate water supply standards of composite char-
acter represented one of the earliest applications of semi-statistical
method. Most of these were based upon the erroneous conclusion
that methods of evaluating units had been standardized throughout
the country. Attention has been called to this fallacy of endeavoring
to establish limiting values of units attained by varying methods by
Hinman 20 , Norton 21 , and Morse and Wolman 22 . Fundamental train-
ing in statisticaHnterpretation no doubt would prevent the adoption
of water supply quality standards before the principles of unit evalua-
tion have been rigidly enforced.

It is not amiss, perhaps, to call attention at this point to the close
analogy between the so-called scoring of a water supply, or the quanti-
tative allocation of the quality upon the scale of sanitary safety, and
the statistician's concept of index numbers. Wolman 14 has shown
recently that the operations involved in making a price index number
are similar to those followed, to a greater or less extent, by investiga-
tors of water supply scores. In the case of price index numbers, the
object of weighing is to give each commodity included in the index
number an influence upon the results corresponding to its commercial
importance. In water supply index numbers, the object of weighing
likewise is to give each factor making up the score an influence upon

59] Statistical Method in Problems of Water Supply Quality 199

the results corresponding to its sanitary importance. Although the
problems in the two fields are the same, their solutions are necessarily-
different, since, in the case of water supply scores, the conversion to a
common base of such units as bacterial results, sanitary surveys, opera-
ting efficiencies, etc., cannot be carried out because of the presence of
varying personal opinion or judgment. It has been noted 14 , however,
that it still remains possible to make use nationally of simplified index
numbers of water supply quality restricted in their range of signifi-
cance and composed of similar units or, better still, of individual units,
provided the method of evaluation of such units has been definitely
and completely fixed.

Interpretations of the quality of a water include frequently more
than a summary of the structural and environmental features of the
supply. The possibilities of the intelligent and fruitful application of
statistical devices, such as the coefficients of correlation and of varia-
tion, to other phases of water supply are mentioned only briefly here,
since their complete discussion would involve a paper of a far too great
length. Whipple* for instance, has suggested the use of the coefficient
of correlation in analyzing the vital statistics of cities which have made
changes from poor to good quality water supplies, in order to demon-
strate quantitatively the existence of the Mills-Reincke phenomenon.
Hazen 15 has made excellent use of statistical method in his analysis of
the storage provided in an impounding reservoir on any stream and
the quantity of water which can be supplied continuously by it. He
introduces the coefficient of variation as a measure of the degree of
variation in flows of different streams and by its further use has found
it possible to get an approximate expression for the storage required to
carry the surplus water of wet years over to dry years, which expres-
sion, in general terms, applied equally well to streams in different
localities. In addition, he describes methods of estimating the proba-
ble errors in the results obtained and makes the important comment
that "frank recognition of the large probable errors in many of the
results cannot fail to be advantageous." 15

The opportunities for further application of similar methods have
appeared in the present writer's studies of the correlation of bacterial
contents in water supplies with rainfalls upon stream watersheds and
with hygienic resultants of inferior quality such as typhoid fever and
diarrhoeal diseases. In these particular studies, the statistician could
contribute excellent aid, since the writer is not aware of an effective
method of comparing correlated phenomena in which one series of
characteristics is continuous, while another is discontinuous. In addi-

* Personal communication.

200 American Statistical Association [60

tion, quantitative variations in magnitude of the values in both series
are not of paramount importance, but the direction of such variations
is the interesting event. The coefficient of concurrent deviations in
this instance, does not appear to supply all the desiderata. An exam-
ple may make our problem clearer. In the study of the daily tap
water analyses of a city water supply, we find, by inspection, that the
B. coli contents rise after rains on the watershed of the stream supply-
ing the town. It is also found that such rises are masked, to varying
degrees, by purification processes and by the efficiency of operation
of such processes. If changes in method and efficiency of purification
are brought about and the qualitative reflection of rainfalls in resultant
B. coli density in tap waters is modified, how can we measure quantita-
tively the change in sensitiveness of tap water quality to rainfall from
month to month? The data at hand for this purpose, reduced to
simplest terms, are in each month B. coli values for each day (continu-
ous series), which differ in density from day to day, and rainfall records
(discontinuous series) which may give a zero value for all the days but
three or four during the month. If, during the month of July, the B.
coli per 100 c.c. rose from 2 to 2,000 from July 7 to July 8, following a
rain of 0.8 inch on the stream on July 7, and during August the B. coli
per 100 c.c. showed no jumps above 5 in spite of a number of days of
rainfall of about 0.8 inch, what should be the statistical relation be-
tween the months of July and August for these particular considerations?

This paper should not be concluded without some reference to the
part that the study of purification processes has played in modifying
and determining the quality of water supplies and the importance
therein of the mathematician's tools. It is frequently the sanitarian's
problem to include in his valuation of a water's safety some definite
estimate, among other things, of the efficiency of operating features
involved in the treatment of such a supply. This problem has given
rise to various measures of treatment efficiencies, which only recently
have been subjected to rigid statistical study. As an illustration of
this type of measure the percentage removal of bacteria from untreated
to treated waters has persisted. Statistical objections to this measure
are well known to the reader and substitutes for this measure of per-
formance, and indirectly of quality, have been much sought after. It
was long recognized that the real measure of performance should in-
clude data regarding the distribution of the efficiencies over long periods
and recommendations suggesting the classification of bacterial results
according to frequency distributions have done much to clarify the
interpretation of treatment figures.

Further development of the same problem of plant performance

61] Statistical Method in Problems of Water Supply Quality 201

along statistical lines has been made by Wolman 23 , in the study of the
nature of bacterial removal in filtration plants. In this discussion, it
was suggested that "the normal performance of a water filtration plant
may be represented by a curve having the equation: y = x c , where
y and x are respectively the raw water and final effluents counts, and c
is a constant for the particular plant under discussion." In other
words, the tentative hypothesis was brought forth that the final efflu-
ent count, on the average, is an exponential function of the raw water
count. The evaluation of "c" replaces also the unsatisfactory per-
centage efficiency as a more adequate measure, by using the ratio of
the logarithms of the counts instead of the ratio of the actual bacterial
values.

It is apparent that a measure of performance to be effective for
adaptation to quality interpretation should include more than an array
of its daily values, since it is the consistency of bacterial removal which
predetermines the position of a form of treatment in the scale of the
safety of a supply. Heretofore, no single unit of measure of this degree
of consistency of removal has been available, although the fitting of
normal performance data to the logarithmic curve of filtration supplied
at least a graphic method of testing consistency. 2 ? If bacterial data
are arranged and plotted on the probability paper already referred to
in the discussion, it becomes extremely easy to obtain the values of the
semi-interquartile ranges of the figures in successive steps of purifica-
tion. The ratio of such values of the ranges for any two steps appears
to the writer to present some promise of a real measure of the "level-
ling" effect of purification processes, since it measures the change pro-
duced in the frequency distribution of bacteria in passing through the
treatment. The demonstration of its value may be more apparent to
the reader by reference to material given elsewhere. 24

REFERENCES

1 Yule, G. U., An Introduction to the Theory of Statistics.

2 Levine, Max, A Statistical Classification of the Colon-Cloacae Group, Journal of

Bacteriology, Vol. 3, No. 3, May, 1918.

3 McCrady, M. H., The Numerical Interpretation of Fermentation-Tube Results,

Journal of Infectious Diseases, Vol. 17, No. 1, July, 1915.

4 Wolman, Abel and Weaver, H. L., A Modification of the McCrady Method of the

Numerical Interpretation of Fermentation-Tube Results, Journal of Infectious
Diseases, Vol. 21, No. 3, September, 1917.

5 McCrady, M. H., Tables for Rapid Interpretation of Fermentation-Tube Results,

The Public Health Journal (Canada), Vol. 9, No. 5, May, 1918.

6 Stein, Milton F., Making the B. Coli Test Tell More, Engineering News-Record,

Vol. 78, No. 8, May 24, 1917.

202 American Statistical Association [62

7 Stein, Milton F., On Numerical Interpretation of Bacteriological Tests, Engineering

News-Record, Vol. 82, No. 23, June 5, 1919.

8 Stein, Milton F., The Interpretation of B. Coli Test Results on a Numerical and

Comparative Basis, Journal of Bacteriology, Vol. 4, No. 3, May, 1919.

9 Greenwood, J. Junr. and Yule, G. U.dny, On The Statistical Interpretation of Some

Bacteriological Methods Employed in Water Analysis, Journal of Hygiene,
Vol. 16, No. 1, July, 1917.

10 Wells, Wm. F., The Geometrical Mean as a B. Coli Index, Science, N. S., Vol. 47,

No. 1202, January 11, 1918.

11 Wells, Wm. F., The Bacteriological Dilution Scale and the Dilution as a Bacterio-

logical Unit, American Journal of Public Health, Vol. 9, No. 9, September,
1919.
n Wells, Wm. F., On a Standard System of Bacteriological Dilutions, American
Journal of Public Health, Vol. 9, No. 12, December, 1919.

13 Whipple, G. C, The Element of Chance in Sanitation, Journal of the Franklin

Institute, Vol. 182, No. 1, No. 2, July and August, 1916.

14 Wolman, Abel, Index Numbers and Scoring of Water Supplies, Journal of the Amer-

ican Water Works Association, Vol. 6, No. 3, September, 1919.

15 Hazen, Allen, Storage to be Provided in Impounding Reservoirs for Municipal

Water Supply, Trans. American Society of Civil Engineers, Vol. 77, p. 1539.

16 Stein, Milton F., A Critical Study of the Bacterial Count in Water and Sewage,

American Journal of Public Health, Vol. 8, No. 11, November, 1918.

17 McEwen, George F. and Michael, Ellis L., The Functional Relation of One Variable

to Each of a Number of Correlated Variables Determined by a Method of
Successive Approximation to Group Averages: A Contribution to Statistical
Methods, Proc. American Academy Arts and Sciences, Vol. 55, No. 2, Decem-
ber, 1919.

18 Michael, Ellis L., Concerning Application of the Probable Error in Cases of

Extremely Asymmetrical Frequency Curves, Science, N. S., Vol. 51, No. 1308,
January 23, 1920.

19 Johnstone, James, The Probable Error of a Bacteriological Analysis, Bept. Lane.

Sea-Fish. Lab., 1919, No. 27. (Not read.)

20 Hinman, J. J. Jr., American Water Works Laboratories, Journal of the American

Water Works Association, Vol. 5, No. 2, June, 1918.

21 Norton, J. F., Comparison of Methods for the Examination of Water at Filtration

Plants, Journal of Infectious Diseases, Vol. 23, 1918, Pp. 344-50.

22 Morse, Robert B. and Wolman, Abel, The Practicability of Adopting Standards of

Quality for Water Supplies, Journal of the American Water Works Association,
Vol. 5, No. 3, September, 1918.

23 Wolman, Abel, A Preliminary Analysis of the Degree and Nature of Bacterial

Removal in Filtration Plants, Journal of The American Water Works Associa-
tion, Vol. 5, No. 3, September, 1918.

24 Wolman, Abel and Powell, S. T., Sanitary Effect of Water Storage in Open Reser-

voirs, Engineering News-Record, Vol. 83, No. 18, October 30-November 6,
1919.

```